Self-Consistent Field Theory Study of the Effect of Grafting Density on the Height of a Weak Polyelectrolyte Brush Kevin N. Witte, Sangtae Kim, and You-Yeon Won* School of Chemical Engineering, Purdue UniVersity, 480 Stadium Mall DriVe, West Lafayette, Indiana 47907 ReceiVed: NoVember 6, 2008; ReVised Manuscript ReceiVed: June 28, 2009 The height of weakly basic polyelectrolyte brushes in the osmotic brush regime is studied as a function of the grafting density using a numerical self-consistent field theory derived from the (semi)grand canonical partition function. The theory is shown to properly account for the local nature of the charge equilibrium and to capture the basic behaviors of polyelectrolyte brushes. On one hand, we find, in agreement with recent experiments, that the scaling of brush height with grafting density can be qualitatively different at intermediate chain lengths than that predicted by basic scaling arguments. This difference is attributed to the relative strength of electrostatic type interactions compared to finite segment size packing constraints. On the other hand, the trend of decreasing brush height with increasing grafting density predicted by the classic scaling analysis is recovered for large molecular weight polymers immersed in a solution of very weak ionic strength. 1. Introduction Weak polyelectrolyte brushes are polymers grafted to a surface or interface at a high enough density that the chains are stretched away from their coiled state and whose monomers contain weakly ionizable groups, such as carboxyl or amine groups. 1-5 These systems have recently received significant attention due to the highly tunable nature of their properties and chain conformations. In particular, the height can be manipulated by varying the pH, salt concentration, or temperature of the bulk (aqueous) solvent. In general, this tunability derives from a chemical equilibrium established by the weakly ionizable side groups with the solvent. Thus, by changing the pH for a weakly acidic or basic polymer, the equilibrium is shifted, and the degree of charging on the chain can be varied. In contrast, a strong polyelectrolyte has a constant degree of charging regardless of external condi- tions. (See ref 4 for a review.) In order to understand the complex behavior of these brush systems, simple scaling theories have been proposed. These theories attempt to balance the forces that tend to form a Gaussian coil (coil elastic energy) with those that tend to swell the brush (osmotic pressure and electrostatic interac- tions). For a strong polyelectrolyte brush it has been shown that the brush height in the osmotic brush regime, which is the regime where the osmotic pressure of the small ions is dominant, scales as H ∼ Nbf 1/2 ; where N is the number of segments of size b, and f is the fraction of segments that are charged. 3,6 Note that the brush height is found to be independent of the ion concentration, C S , and the grafting density of the polymers, σ. For weak polyelectrolytes, on the other hand, the brush height scales as H ∼Nb 4/3 σ -1/3 C S 1/3 . Here the brush height is expected to increase with salt concentration throughout the osmotic regime due to the replacement of counterions (e.g., H + ) with salt ions (e.g., Na + ) that cannot participate to any appreciable extent in the equilibrium. Thus, the reaction is driven toward the products resulting in more charges along the chain and a more swollen polyacid chain. 3 The brush height is expected to decrease with an increase in the grafting density due to the greater concentration of monomers driving the equilibrium toward the reactants. Thus, the average degree of charging will be decreased, and the brush will collapse relative to a lower grafting density. Using a lattice self-consistent field (SCF) theory, in the high molecular weight and low salt limits, Israe ¨ls et al. found qualitatively excellent agreement with the prediction that brush height should decrease with an increase in grafting density within the osmotic brush regime. However, they noted, in the low salt regions of the osmotic brush regime, the SCF theory gave larger brush heights than expected from the scaling theory. They attributed these deviations to “excluded volume interactions” not accounted for in the scaling predictions. 1 Recent experiments by Genzer and co-workers have shown that the predicted scaling relations do not necessarily hold throughout the entire range of the osmotic brush regime. 7 Using a “grafting from” approach, poly(acrylic acid) (PAA) brushes with approximately 70 acrylic acid residues were grown on a solid substrate in such a way that a grafting point density gradient was created. The brush height was measured using elipsometry as a function of the grafting density for various values of solution pH and ionic strength. The authors found that, regardless of the salt concentration, the PAA brush height increased with grafting density to approximately the +1/3 power. In a concurrent paper, the experimental results were scaled at each grafting density and compared to the results of a single chain mean field (SCMF) theory. 8 Good agreement was found for the brush heights as a function of salt concentration at a wide variety of system conditions. The authors were then able to explain the observed height variations in terms of the local field variables. They unfortunately did not explicitly comment on why the brush height increases with grafting density in contrast to the predicted scaling relation. Similar trends can also be observed in the results of Currie et al. 9 However, these authors used a system of polystyrene- poly(acrylic acid) (PS-PAA) spread at the air-water interface * Corresponding author. E-mail: yywon@ecn.purdue.edu. J. Phys. Chem. B 2009, 113, 11076–11084 11076 10.1021/jp809814j CCC: $40.75  2009 American Chemical Society Published on Web 07/17/2009 Downloaded by PURDUE UNIV on August 23, 2009 Published on July 17, 2009 on http://pubs.acs.org | doi: 10.1021/jp809814j